Forjustastheuniversalpropositionsofmathematicsdealnot
withobjectswhichexistseparately，apartfromextendedmagnitudes
andfromnumbers，butwithmagnitudesandnumbers，nothoweverqua
suchastohavemagnitudeortobedivisible，clearlyitispossible
thatthereshouldalsobebothpropositionsanddemonstrationsabout
sensiblemagnitudes，nothoweverquasensiblebutquapossessedof
certaindefinitequalities。Forastherearemanypropositionsabout
thingsmerelyconsideredasinmotion，apartfromwhateachsuchthing
isandfromtheiraccidents，andasitisnotthereforenecessarythat
thereshouldbeeitheramobileseparatefromsensibles，oradistinct
mobileentityinthesensibles，sotoointhecaseofmobilesthere
willbepropositionsandsciences，whichtreatthemhowevernotqua
mobilebutonlyquabodies，oragainonlyquaplanes，oronlyqua
lines，orquadivisibles，orquaindivisibleshavingposition，oronly
quaindivisibles。Thussinceitistruetosaywithoutqualification
thatnotonlythingswhichareseparablebutalsothingswhichare
inseparableexist(forinstance，thatmobilesexist)，itistrue
alsotosaywithoutqualificationthattheobjectsofmathematics
exist，andwiththecharacterascribedtothembymathematicians。
Andasitistruetosayoftheothersciencestoo，without
qualification，thattheydealwithsuchandsuchasubject-notwith
whatisaccidentaltoit(e。g。notwiththepale，ifthehealthything
ispale，andthesciencehasthehealthyasitssubject)，butwith
thatwhichisthesubjectofeachscience-withthehealthyifit
treatsitsobjectquahealthy，withmanifquaman:-sotooisit
withgeometry；ifitssubjectshappentobesensible，thoughitdoes
nottreatthemquasensible，themathematicalscienceswillnotfor
thatreasonbesciencesofsensibles-nor，ontheotherhand，of
otherthingsseparatefromsensibles。Manypropertiesattachtothings
invirtueoftheirownnatureaspossessedofeachsuchcharacter；
e。g。thereareattributespeculiartotheanimalquafemaleorqua
male(yetthereisno’female’nor’male’separatefromanimals)；so
thattherearealsoattributeswhichbelongtothingsmerelyas
lengthsorasplanes。Andinproportionaswearedealingwith
thingswhicharepriorindefinitionandsimpler，ourknowledgehas
moreaccuracy，i。e。simplicity。Thereforeasciencewhichabstracts
fromspatialmagnitudeismoreprecisethanonewhichtakesitinto
account；andascienceismostpreciseifitabstractsfrom
movement，butifittakesaccountofmovement，itismostpreciseif
itdealswiththeprimarymovement，forthisisthesimplest；andof
thisagainuniformmovementisthesimplestform。
Thesameaccountmaybegivenofharmonicsandoptics；forneither
considersitsobjectsquasightorquavoice，butqualinesand
numbers；butthelatterareattributespropertotheformer。And
mechanicstooproceedsinthesameway。Thereforeifwesuppose
attributesseparatedfromtheirfellowattributesandmakeanyinquiry
concerningthemassuch，weshallnotforthisreasonbeinerror，any
morethanwhenonedrawsalineonthegroundandcallsitafootlong
whenitisnot；fortheerrorisnotincludedinthepremisses。
Eachquestionwillbebestinvestigatedinthisway-bysetting
upbyanactofseparationwhatisnotseparate，asthe
arithmeticianandthegeometerdo。Foramanquamanisone
indivisiblething；andthearithmeticiansupposedoneindivisible
thing，andthenconsideredwhetheranyattributebelongstoaman
quaindivisible。Butthegeometertreatshimneitherquamannorqua
indivisible，butasasolid。Forevidentlythepropertieswhich
wouldhavebelongedtohimevenifperchancehehadnotbeen
indivisible，canbelongtohimevenapartfromtheseattributes。Thus，
then，geometersspeakcorrectly；theytalkaboutexistingthings，
andtheirsubjectsdoexist；forbeinghastwoforms-itexistsnot
onlyincompleterealitybutalsomaterially。
Nowsincethegoodandthebeautifularedifferent(fortheformer
alwaysimpliesconductasitssubject，whilethebeautifulisfound
alsoinmotionlessthings)，thosewhoassertthatthemathematical
sciencessaynothingofthebeautifulorthegoodareinerror。For
thesesciencessayandproveagreatdealaboutthem；iftheydonot
expresslymentionthem，butproveattributeswhicharetheirresults
ortheirdefinitions，itisnottruetosaythattheytellus
nothingaboutthem。Thechiefformsofbeautyareorderandsymmetry
anddefiniteness，whichthemathematicalsciencesdemonstrateina
specialdegree。Andsincethese(e。g。orderanddefiniteness)are
obviouslycausesofmanythings，evidentlythesesciencesmusttreat
thissortofcausativeprinciplealso(i。e。thebeautiful)asin
somesenseacause。Butweshallspeakmoreplainlyelsewhereabout
thesematters。
Somuchthenfortheobjectsofmathematics；wehavesaidthat
theyexistandinwhatsensetheyexist，andinwhatsensetheyare
priorandinwhatsensenotprior。Now，regardingtheIdeas，wemust
firstexaminetheidealtheoryitself，notconnectingitinanyway
withthenatureofnumbers，buttreatingitintheforminwhichit
wasoriginallyunderstoodbythosewhofirstmaintainedthe
existenceoftheIdeas。Thesupportersoftheidealtheorywereledto
itbecauseonthequestionaboutthetruthofthingstheyacceptedthe
Heracliteansayingswhichdescribeallsensiblethingsaseverpassing
away，sothatifknowledgeorthoughtistohaveanobject，theremust
besomeotherandpermanententities，apartfromthosewhichare
sensible；fortherecouldbenoknowledgeofthingswhichwereina
stateofflux。ButwhenSocrateswasoccupyinghimselfwiththe
excellencesofcharacter，andinconnexionwiththembecamethe
firsttoraisetheproblemofuniversaldefinition(forofthe
physicistsDemocritusonlytouchedonthesubjecttoasmallextent，
anddefined，afterafashion，thehotandthecold；whilethe
Pythagoreanshadbeforethistreatedofafewthings，whose
definitions-e。g。thoseofopportunity，justice，ormarriage-they
connectedwithnumbers；butitwasnaturalthatSocratesshouldbe
seekingtheessence，forhewasseekingtosyllogize，and’whata
thingis’isthestarting-pointofsyllogisms；fortherewasasyet
noneofthedialecticalpowerwhichenablespeopleevenwithout
knowledgeoftheessencetospeculateaboutcontrariesandinquire
whetherthesamesciencedealswithcontraries；fortwothingsmay
befairlyascribedtoSocrates-inductiveargumentsanduniversal
definition，bothofwhichareconcernedwiththestarting-pointof
science):-butSocratesdidnotmaketheuniversalsorthe
definitionsexistapart:they，however，gavethemseparate
existence，andthiswasthekindofthingtheycalledIdeas。Therefore
itfollowedforthem，almostbythesameargument，thattheremust
beIdeasofallthingsthatarespokenofuniversally，anditwas
almostasifamanwishedtocountcertainthings，andwhiletheywere
fewthoughthewouldnotbeabletocountthem，butmademoreof
themandthencountedthem；fortheFormsare，onemaysay，more
numerousthantheparticularsensiblethings，yetitwasinseeking
thecausesofthesethattheyproceededfromthemtotheForms。Forto
eachthingthereanswersanentitywhichhasthesamenameand
existsapartfromthesubstances，andsoalsointhecaseofallother
groupsthereisaoneovermany，whetherthesebeofthisworldor
eternal。
Again，ofthewaysinwhichitisprovedthattheFormsexist，
noneisconvincing；forfromsomenoinferencenecessarilyfollows，
andfromsomeariseFormsevenofthingsofwhichtheythinkthereare
noForms。Foraccordingtotheargumentsfromthesciencesthere
willbeFormsofallthingsofwhichtherearesciences，andaccording
totheargumentofthe’oneovermany’therewillbeFormsevenof
negations，andaccordingtotheargumentthatthoughthasanobject
whentheindividualobjecthasperished，therewillbeFormsof
perishablethings；forwehaveanimageofthese。Again，ofthemost
accuratearguments，someleadtoIdeasofrelations，ofwhichtheysay
thereisnoindependentclass，andothersintroducethe’thirdman’。
AndingeneraltheargumentsfortheFormsdestroythingsfor
whoseexistencethebelieversinFormsaremorezealousthanforthe
existenceoftheIdeas；foritfollowsthatnotthedyadbutnumberis
first，andthatpriortonumberistherelative，andthatthisis
priortotheabsolute-besidesalltheotherpointsonwhichcertain
people，byfollowingouttheopinionsheldabouttheForms，came
intoconflictwiththeprinciplesofthetheory。
Again，accordingtotheassumptiononthebeliefintheIdeas
rests，therewillbeFormsnotonlyofsubstancesbutalsoofmany
otherthings；fortheconceptissinglenotonlyinthecaseof
substances，butalsointhatofnon-substances，andtherearesciences
ofotherthingsthansubstance；andathousandothersuchdifficulties
confrontthem。Butaccordingtothenecessitiesofthecaseandthe
opinionsabouttheForms，iftheycanbesharedintheremustbeIdeas
ofsubstancesonly。Fortheyarenotsharedinincidentally，but
eachFormmustbesharedinassomethingnotpredicatedofa
subject。(By’beingsharedinincidentally’Imeanthatifathing
sharesin’doubleitself’，itsharesalsoin’eternal’，but
incidentally；for’thedouble’happenstobeeternal。)Thereforethe
Formswillbesubstance。Butthesamenamesindicatesubstanceinthis
andintheidealworld(orwhatwillbethemeaningofsayingthat
thereissomethingapartfromtheparticulars-theoneovermany？)。And
iftheIdeasandthethingsthatshareinthemhavethesameform，
therewillbesomethingcommon:forwhyshould’2’beoneandthesame
intheperishable2’s，orinthe2’swhicharemanybuteternal，and
notthesameinthe’2itself’asintheindividual2？Butifthey
havenotthesameform，theywillhaveonlythenameincommon，andit
isasifoneweretocallbothCalliasandapieceofwooda’man’，
withoutobservinganycommunitybetweenthem。
Butifwearetosupposethatinotherrespectsthecommon
definitionsapplytotheForms，e。g。that’planefigure’andtheother
partsofthedefinitionapplytothecircleitself，but’whatreally
is’hastobeadded，wemustinquirewhetherthisisnotabsolutely
meaningless。Fortowhatisthistobeadded？To’centre’orto
’plane’ortoallthepartsofthedefinition？Foralltheelementsin
theessenceareIdeas，e。g。’animal’and’two-footed’。Further，
theremustbesomeIdealansweringto’plane’above，somenaturewhich
willbepresentinalltheFormsastheirgenus。
Aboveallonemightdiscussthequestionwhatintheworldthe
Formscontributetosensiblethings，eithertothosethatare
eternalortothosethatcomeintobeingandceasetobe；forthey
causeneithermovementnoranychangeinthem。Butagaintheyhelp
innowiseeithertowardstheknowledgeofotherthings(forthey
arenoteventhesubstanceofthese，elsetheywouldhavebeenin
them)，ortowardstheirbeing，iftheyarenotintheindividuals
whichshareinthem；thoughiftheywere，theymightbethoughtto
becauses，aswhitecauseswhitenessinawhiteobjectbyentering
intoitscomposition。Butthisargument，whichwasusedfirstby
Anaxagoras，andlaterbyEudoxusinhisdiscussionofdifficultiesand
bycertainothers，isveryeasilyupset；foritiseasytocollect
manyandinsuperableobjectionstosuchaview。
